A Canonical Partition Theorem for Equivalence Relations on Zt

THEOREM A. For every positive integer m there exists a positive integer n such that for every coloring A: (O,..., n 1) + w there exists an arithmetic progression a, a + d,..., a + (m 1)d of length m such that the restriction of A to {a, a + d,..., a + (m l)d} is either a constant or a one-to-one mapping. This theorem is the so called canonical version of van der Waerden’s theorem on arithmetic progressions [7]. Originally, the consideration of canonical partition theorems goes back to Erdiis and Rado [2 1 who proved this generalization of Ransey’s theorem known as “Erd6s-Rado canonization theorem.”